Volume 4 • Number 2 • 2017 ISSN 2312-4334

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(1)Volume 4 • Number 2 • 2017. V.N. Karazin Kharkiv National University Publishing. ISSN 2312-4334.

(2) ISSN 2312-4334 MINISTRY OF EDUCATION AND SCIENCE OF UKRAINE. Volume 4 · Number 2. 2017.

(3) East European Journal of Physics EEJP is an international peer-reviewed journal devoted to experimental and theoretical research on the nuclear physics, cosmic rays and particles, high-energy physics, solid state physics, plasma physics, physics of charged particle beams, plasma electronics, radiation materials science, physics of thin films, condensed matter physics, functional materials and coatings, medical physics and physical technologies in an interdisciplinary context. Published quarterly in hard copy and online by V.N. Karazin Kharkiv National University Publishing. ISSN 2312-4334 (Print), ISSN 2312-4539 (Online) The editorial policy is to maintain the quality of published papers at the highest level by strict peer review. Approved for publication by the Academic Council of the Karazin Kharkiv National University (May 29, 2017, Protocol No. 7). EEJP registered by the order of Ministry of Education of Ukraine № 747 of 07.13.2015, and is included in the list of scientific specialized editions of Ukraine, which can be published results of dissertations for the degree of doctor and candidate of physical and mathematical sciences. Editor-in-Chief Azarenkov N.A., Karazin Kharkiv National University, Kharkiv, Ukraine Deputy editor Girka I.O., Karazin Kharkiv National University, Kharkiv, Ukraine Executive Secretary Girnyk S.A., Karazin Kharkiv National University, Kharkiv, Ukraine Editorial Board Adamenko I.N., Karazin Kharkiv National University, Ukraine Akulov V.P., City University of New York, USA Antonov A.N., Institute of Nuclear Research and Nuclear Energy, Sofia, Bulgaria Barannik E.A., Karazin Kharkiv National University, Ukraine Beresnev V.M., Karazin Kharkiv National University, Ukraine Berezhnoy Yu.A., Karazin Kharkiv National University, Ukraine Bizyukov A.A., Karazin Kharkiv National University, Ukraine Bragina L.L. STU Kharkiv Polytechnical Institute, Ukraine Broda B., University of Lodz, Poland Budagov Yu.A., Joint Institute of Nuclear Research, Dubna, Russia Dovbnya A.M., NSC Kharkiv Institute of Physics and Technology, Ukraine Dragovich B.G., University of Belgrade, Serbia Duplij S.A., Karazin Kharkiv National University, Ukraine Garkusha I.E., NSC Kharkiv Institute of Physics and Technology, Ukraine Gofman Yu., Jerusalem College of Technology, Israel Grekov D.L., NSC Kharkiv Institute of Physics and Technology, Ukraine Karnaukhov I.M., NSC Kharkiv Institute of Physics and Technology, Ukraine Khodusov V.D., Karazin Kharkiv National University, Ukraine Kondratenko A.N., Karazin Kharkiv National University, Ukraine Korchin A.Yu., NSC Kharkiv Institute of Physics and Technology, Ukraine Krivoruchenko M.I., Institute for Theoretical and Experimental Physics, Moscow, Russia Lavrinenko S.D., NSC Kharkiv Institute of Physics and Technology, Ukraine Lazurik V.T., Karazin Kharkiv National University, Ukraine Mel'nik V.N., Institute of Radio Astronomy, Kharkiv, Ukraine Merenkov N.P., NSC Kharkiv Institute of Physics and Technology, Ukraine Neklyudov І.М., NSC Kharkiv Institute of Physics and Technology, Ukraine Noterdaeme J.-M., Max Planck Institute for Plasma Physics, Garching, Germany Nurmagambetov A.Yu., NSC Kharkiv Institute of Physics and Technology, Ukraine Onyschenko I.M., NSC Kharkiv Institute of Physics and Technology, Ukraine Ostrikov K.N., Plasma Nanoscience Centre Australia, Clayton, Australia Peletminsky S.V., NSC Kharkiv Institute of Physics and Technology, Ukraine Pilipenko N.N., NSC Kharkiv Institute of Physics and Technology, Ukraine Radinschi I., Gheorghe Asachi Technical University, Iasi, Romania Slyusarenko Yu.V., NSC Kharkiv Institute of Physics and Technology, Ukraine Smolyakov А.I., University of Saskatchewan, Saskatoon, Canada Shul'ga N.F., NSC Kharkiv Institute of Physics and Technology, Ukraine Tkachenko V.I., NSC Kharkiv Institute of Physics and Technology, Ukraine Voyevodin V.M., NSC Kharkiv Institute of Physics and Technology, Ukraine Yegorov O.M., NSC Kharkiv Institute of Physics and Technology, Ukraine Editorial office Department of Physics and Technologies, V.N. Karazin Kharkiv National University Kurchatov av., 31, office 402, Kharkiv, 61108, Ukraine Tel: +38-057-335-18-33, E-mail: [email protected], Web-pages: http://periodicals.karazin.ua/eejp (Open Journal System), http://eejp.univer.kharkov.ua Certificate of State registration No.20644-10464P, 21.02.2014. ©. V. N. Karazin Kharkiv National University Publishing, design, 2017.

(4) East European Journal of Physics Volume 4. Number 2. 2017. ORIGINAL PAPERS Models of Hamiltonian and Low-Frequency Spectra of Collective Excitations in Spin S = 1 Magnetics A.V. Glushchenko, M.Y. Kovalevsky. 4. Increase of Amplitude of Accelerating Wakefield, Excited by Sequence of Short Relativistic Electron Bunches in Plasma, at Magnetic Field Use D.S. Bondar, I.P. Levchuk, V.I. Maslov, I.N. Onishchenko. 11. Effect of Amyloid Fibrils on Electrokinetic Properties of Lipid Vesicles U. Tarabara, K. Vus, S. Girnyk, N. Kamneva, O. Lavryk, M. Mikhailyuta, V. Trusova, G. Gorbenko. 19. Dynamics of Bose-Einstein Condensate with Account of Pair Correlations Yu.M. Poluektov, A.M. Arslanaliev. 29. Clinoptilolite with Cesium Immobilisation to Potassium Magnesium Phosphate Matrix S.Yu. Sayenko, V.A. Shkuropatenko, N.P. Dikiy, R.V. Tarasov, K.A. Ulybkina, O.Y. Surkov, L.M. Litvinenko. 37. Low Temperature Deformation and Strength of Polyimide Films Due to Thickness and Deformation Speed V.A. Lototskaya, L.F. Yakovenko, E.N. Aleksenko, V.V. Abraimov, Wen Zhu Shao. 44. Measurement of the Main Characteristics of PuBe and 238PuBe Neutron Sources Using Bonner Sphere Spectrometer With A 3He-Counter V.B. Ivanskyi, O.N. Letuchyy, А.N. Orobinskyi, H.V. Siroko. 53. Study of Plastic Deformation of Cadmium I.I. Papirov, P.I. Stoev, G.P. Kovtun, A.P. Shcherban, D.A. Solopikhin, T.Yr. Rudycheva. 66. Maximum Temperature in an Ideal Greenhouse O. Ivashtenko. 78.

(5) 4. EAST EUROPEAN JOURNAL OF PHYSICS. East Eur. J. Phys. Vol.4 No.2 (2017) 4-10. DOI:10.26565/2312-4334-2017-2-01. PACS: 75.10.–b. MODELS OF HAMILTONIAN AND LOW-FREQUENCY SPECTRA OF COLLECTIVE EXCITATIONS IN SPIN S = 1 MAGNETICS A.V. Glushchenko, M.Y. Kovalevsky Kharkov Institute of Physics and Technology Academicheskaya 1, Kharkov, 61108, Ukraine E-mail: [email protected] Received March 27, 2017 The paper studies the dynamic description of non-equilibrium processes in single-sublattice and multisublattice magnets with the spin s=1. In case of magnets with the spin s=1 and SU(3) symmetry of the exchange interaction, there are eight magnetic integrals of motion: the spin and the quadrupole matrix. If there are multiple sublattices, the number of additional magnetic quantities characterizing the state increases to sixteen. The presence of the Casimir invariants makes it possible to reduce the number of independent degrees of freedom. Exchange energy models are presented in terms of Casimir invariants corresponding to SO(3) or SU(3) symmetry groups for all four types of magnetic degrees of freedom. For the homogeneous part of the exchange energy, we have found conditions for the existence of local minima, which correspond to equilibrium values of the magnet. Along with the known waves (quadrupole and Goldstone – for the spin nematic), spectra of collective excitations that take into account ferroquadrupole excitation, quadro-nematic, quadro- antiferromagnetic, and antiferro-nematic waves excitation, are also obtained. In the case of many-sublattice magnetic systems, we have shown that the selected form of the homogeneous energy model allows us to find possible magnetic orderings and to investigate them for stability. KEY WORDS:SU(3) symmetry, magnet, spin, exchange interaction, Casimir invariant, spectra. МОДЕЛІ ГАМИЛЬТОНИАНА І НИЗЬКОЧАСТОТНІ СПЕКТРИ КОЛЕКТИВНИХ ЗБУДЖЕНЬ У МАГНЕТИКАХ ЗІ СПИНОМ S = 1 А.В. Глущенко, М.Ю. Ковалевський Національний науковий центр «Харківський фізико-технічний інститут» вул. Академічна, 1, м Харків, 61108, Україна В роботі розглянуті питання динамічного опису нерівноважних процесів в однопідгратковому та багатопідграткових магнетиках зі спіном s = 1. У разі магнетиків зі спіном s = 1 і SU (3) симетрії обмінної взаємодії магнітних інтегралів руху вісім: це спін і квадрупольна матриця. За наявності кількох підграток, число магнітних величин, що характеризують стан рівноваги, збільшується до шістнадцяти. Наявність інваріантів Казимира дозволяє зменшити число незалежних ступенів свободи. Моделі обмінної енергії представлені в термінах інваріантів Казимира, що відповідають групам SO(3) або SU(3) симетрії, для всіх чотирьох типів магнітних ступенів свободи. Для однорідної частини обмінної енергії знайдені умови існування локальних мінімумів, які відповідають рівноважним значенням магнетика. Поряд з відомими хвилями (квадрупольними і голдстоунівськими для спінового нематика), також отримані іншого виду спектри колективних збуджень, які описують феро-квадрупольне збудження, а також квадро-нематичні, квадро-антиферомагнітні і антіферронематичні хвилі. Нами показано, що у разі багатопідграткових магнітних систем вид однорідної моделі енергії дозволяє знайти можливі магнітні впорядкування і досліджувати їх на стійкість. КЛЮЧОВІ СЛОВА: SU(3) симетрія, магнетик, спін, обмінна взаємодія, інваріант Казимира, спектри МОДЕЛИ ГАМИЛЬТОНИАНА И НИЗКОЧАСТОТНЫЕ СПЕКТРЫ КОЛЛЕКТИВНЫХ ВОЗБУЖДЕНИЙ В МАГНЕТИКАХ СО СПИНОМ S=1 А.В. Глущенко, М.Ю. Ковалевский Национальный научный центр «Харьковский физико-технический институт» ул. Академическая, 1, г. Харьков, 61108, Украина В работе рассмотрены вопросы динамического описания неравновесных процессов в одноподрешеточном и многоподрешеточном магнетиках со спином s=1. В случае магнетиков со спином s=1 и SU(3) симметрии обменного взаимодействия магнитных интегралов движения восемь: это спин и квадрупольная матрица. Если имеется несколько подрешеток, то число магнитных величин, характеризующих состояние, увеличивается до шестнадцати. Наличие инвариантов Казимира позволяет уменьшить число независимых степеней свободы. Модели обменной энергии представлены в терминах инвариантов Казимира, отвечающих группам SO(3) или SU(3) симметрии, для всех четырех типов магнитных степеней свободы. Для однородной части обменной энергии найдены условия существования локальных минимумов, которые отвечают равновесным значениям магнетика. Наряду с известными волнами (квадрупольная и голдстоуновская для спинового нематика), также получены иного вида спектры коллективных возбуждений, которые описывают ферро-квадрупольное возбуждение, а также квадро-нематические, квадро-антиферромагнитные и антиферронематические волны. Нами показано, что в случае многоподрешеточных магнитных систем выбранный вид однородной модели энергии позволяет найти возможные магнитные упорядочения и исследовать их на устойчивость. КЛЮЧЕВЫЕ СЛОВА: SU(3) симметрия, магнетик, спин, обменное взаимодействие, инвариант Казимира, спектры. The description of collective properties of magnets with the spin of the structural element of the medium s>1/2 is of great physical interest due to the emergence of new magnetic states and their expected practical application. The © GlushchenkoA.V. , KovalevskyM.Y., 2017.

(6) 5. Models of Hamiltonian and Low-Frequency Spectra of Collective Excitations.... EEJP Vol.4 No.2 2017. experimental observation of the quadrupole phase [1-3] and spin nematic phase [4] gave a new impetus to the interest in this area of magnetism. Theoretical studies of such multiparticle objects use different concepts and approaches. In the description of the basic magnetic state, a bilinear-biquadratic Hamiltonian [5-8], which is the functional of site sublattice spins, is widely used. In the description of magnets with the spin s=1 and SU(3) symmetry, another approach is possible, in which the quantity in terms of which the Hamilton operator is possible, is the site generator of this symmetry, represented in the Weyl or Racah basis [9,10]. The derivation of dynamic equations of magnets with the spin s=1 for the single-sublattice case in the Racah basis and its relation to the bilinear-biquadratic Hamiltonian are traced in [8,11]. In [12], dynamic equations were obtained in the Weyl basis, both for the single-sublattice and multisublattice magnets with the spin s=1. The relationship between both bases and corresponding equations is considered in [13]. In the study of low-frequency phenomena in magnets, the idea of a spontaneous breaking of a symmetry of the statistical equilibrium state is quite seminal [14]. Using an analogy between magnetic systems of the “easy plane” type and superfluid liquids, an approach was proposed in [15] that made it possible to formulate equations of motion for uniaxial magnets with spontaneous symmetry breaking with respect to spin rotations around the anisotropy axis. In the case of multisublattice magnets with the spin s=1/2, the idea of spontaneous symmetry breaking in the phenomenological approach was used in [16-18]. From the point of view of the symmetry phenomenological approach, any magnetic structure of magnets with the spin s=1/2 can be characterized by no more than six degrees of freedom. Three of them can be conveniently chosen in the form of the spin density, and the remaining three quantities have the physical meaning of the order parameter. We note that only those degrees of freedom of the magnet, which slowly vary in space, are essential in the dynamic description of the low-frequency case. Therefore, the use of the Landau-Lifshitz equation in the low-frequency case for multisublattice magnets is poorly justified, since spins of sublattices are not approximate integrals of motion due to a strong inter lattice exchange interaction. The number of macroscopic magnetic degrees of freedom in this approach is not directly related to the number of sublattices. An essential role belongs to symmetry considerations of the exchange interaction, the equilibrium state, and the residual symmetry of the equilibrium state in the sense of the concept of quasi-averages. In case of magnets with the spin s=1 and SU (3) symmetry of the exchange interaction, there are eight magnetic integrals of motion: the spin and the quadrupole matrix. If there are multiple sublattices, the number of additional magnetic quantities characterizing the state increases. In case of a complete symmetry breaking of the equilibrium state, their number does not exceed eight parameters. Therefore, the total number of magnetic degrees of freedom does not exceed sixteen. Due to the complexity of the magnetic object studied, we use the continuum approximation, in which there are no site spins. Purpose of this paper is to describe the basic state of the magnet with spin s = 1 in the case of one or more sublattices, as well as to study the explicit form of the exchange energy model constructed from Casimir invariants for the Poisson bracket algebra of magnetic degrees of freedom corresponding to the SU(3) and SO(3) symmetry of the interaction and to find the spectra of collective excitations, near the ferro-quadrupole state, quad-nematic, antiferromagnetic and antiferro-nematic states. DEGREES OF FREEDOM IN MAGNETS WITH THE SPIN S=1 In accordance with the approach [12], in order to construct the Hamiltonian mechanics of magnets with the spin   s=1, we introduce Hermitian 3 3 matrices b and a - (aˆ  aˆ , bˆ  bˆ ) , which are canonically conjugate quantities. This means that the following Poisson brackets are valid: b x, b x  0 , a x, a x  0 , b x, a x   x  x . (1). . . . . . . We connect these matrices with physical variables, which are required for constructing the dynamics of magnets with the spin s=1. To this end, we introduce the Hermitian and traceless matrix. . . gˆ x  i bˆx, aˆ x .. (2). This quantity has the physical meaning of the SU(3) symmetry generator density. Using the definition (2) and formula (1), we find the Poisson brackets for this matrix:. .  . . i g  x, g  x  g  x  g  x  x  x.. (3). Formulas (1),(2) allow us to obtain Poisson brackets for matrices aˆ x  and gˆ x  .. .  . . i a x, g  x  a x  a x  x  x.. (4). It is easily seen that Poisson brackets (1),(3),(4) are compatible with the Hermitian requirements of matrices aˆ x  ,. . . gˆ x  and satisfy the Jacobi identities. We note that due to (4), the equality Spaˆ x, g  x  0 is valid. Therefore,.

(7) 6. EEJP Vol.4 No.2 2017. A.V. Glushchenko, M.Y. Kovalevsky. without loss of generality, in view of the linearity of the right-hand side of (4), we can assume that Spaˆ x  0 . Matrices gˆ x  and aˆ x  represent the complete set of magnetic degrees of freedom of magnets with the spin s=1. We introduce. . real magnetic degrees of freedom. They are the spin vector s x and the quadrupole matrix qˆ x  related to the matrix gˆ x  by the relations. . . . . .  . . s  i  g   g , q  g   g / 2  q e e   / 3  q' f  f   / 3 .. (5). Here, q and q  are the modules of the quadrupole matrix, the vectors d  , e , f   d  e form an orthonormal frame.. . For vectors s x , due to (3), (5), the following Poisson bracket is valid:. s x, s x' x  x's x .. (6). Similarly, we find the relations. qx, q x' x  x's x       / 4 .. (7). The physical state of magnets with one sublattice is characterized only by the matrix gˆ x  . In case of an arbitrary number of magnetic sublattices, the physical state is described by both matrices gˆ x  and aˆ x  . We connect the matrix. aˆ x  with real quantities by the relation a x  m x  i  n x / 2 . The vector n here has the physical meaning. of the order parameter of the antiferromagnetism vector. The matrix mˆ x  is symmetric and traceless. This quantity is. .  . . the order parameter of the spin nematic, which we parametrize by the relation m  m k k    / 3  m l l    / 3 . Here m and m are modules of the matrix mˆ , vectors o , k , l  o  k  form an orthonormal frame. For order parameters, we have obtained Poisson brackets with a quadrupole matrix and a spin vector:. s x, n x' x  x'n x , n x, q x' x  x'm x  m x , s x, m x' x  x'm x  m x ,. (8) (9) (10). mx, q x' x  x'n x         / 4 .. (11). The complete set of magnetic degrees of freedom of magnets with the spin s=1 contains quantities of two types that differ in transformational properties with respect to the time reversal operation. In transformations of the reflection of time T, the antiferromagnet and spin vectors change signs: Tn  n , Ts  s . The quadrupole matrix and the order ˆ  mˆ , Tqˆ  qˆ . parameter of the spin nematic do not change during at time reflection operation: Tm Formulas (6)-(11) allow us to identify subalgebras of Poisson brackets and establish the dynamics of magnets with the spin s=1 for various cases of magnetic ordering. Case 1: the minimal subalgebra contains only the spin vector. The use of the Hamiltonian formalism and Poisson brackets (6) leads to the dynamic Landau-Lifshitz theory [19] for the spin s=1/2. Case 2: Poisson brackets (6),(7) allow us to describe the dynamics of normal states of multisublattice magnets and states of single-sublattice magnets with the SU(3) symmetric exchange Hamiltonian. Case 3: the set of magnetic dynamical quantities consists of the spin density sx  and the antiferromagnet vector nx  . Poisson brackets (6),(8) form a closed subalgebra of Poisson brackets and describe the dynamics of an antiferromagnet or ferrimagnet [20]. Case 4: the spin vector sx  and the tensor order parameter mˆ x  form a closed subalgebra of Poisson brackets (6),(10). In this case, the Hamiltonian has the exchange SO(3) symmetry. The T-even spontaneous symmetry breaking of the equilibrium state describes spin nematic states. Magnets with the spin s=1/2 do not possess such magnetic ordering. The dynamics for such magnets has been studied in detail in [21]. Case 5: the set of magnetic dynamical quantities consists of matrices gˆ x  and aˆ x  . This general case corresponds to the complete spontaneous breaking of the SU(3) symmetry of the equilibrium state. Casimir invariants of the Poisson bracket algebra (3) satisfy the relations. g  x  , g  x '  0 , n. . g 2  Spgˆ 2 ,.

(8) 7. Models of Hamiltonian and Low-Frequency Spectra of Collective Excitations.... EEJP Vol.4 No.2 2017. g 3  Spgˆ 3 . The presence of such invariants reduces the number of independent magnetic degrees of freedom to six in case of normal multisublattice and degenerate single-sublattice magnets with the spin s=1. Multisublattice magnets are generally described by sixteen magnetic degrees of freedom. The Poisson bracket algebra (6)-(11) contains Casimir invariants: a2  Spaˆ 2 , a3  Spaˆ 3 , Spgˆaˆ , Spgˆaˆ 2 . Therefore, the number of magnetic independent degrees of freedom decreases to twelve. We note that quantities g 2  Spgˆ 2 and g 3  Spgˆ 3 are Casimir invariants for the Poisson bracket algebra (3). However, for the extended algebra (6)-(11), these quantities are not invariants of this kind due to the relation g n x, a x  0 . In accordance with the definition of [22], for the Poisson. . . bracket algebra (6)-(11), the quantities g 2 and g 3 are called semi-Casimirs. Exchange energy model for the normal and degenerate states In magnets with the spin s=1, there are several possibilities for dynamic behavior with a different set of abbreviated description parameters. The set of these parameters essentially depends on the Hamiltonian and equilibrium state symmetries, which generally may not coincide. While choosing exchange energy models, for simplicity we consider only cases of a uniaxial quadrupole matrix qˆ  q e e   / 3 and uniaxial order parameter of the spin. . . . . ˆ  m f  f   / 3 . Let us consider a single-sublattice magnet. In this case, the Hamiltonian is a density nematic m. functional of the SU(3) symmetry generator H gˆ  . The expression of the homogeneous part of the exchange energy density may be presented as follows: 2. 1  1  A C 2 2 e0   J 0  q 2  s 2   B  q 2  s 2   s 2  s 4 . 2  2  2 4 3 3 Here, g 2  2q 2 / 3  s 2 / 2 . In the energy density, the Casimir invariant. (14). g 2 is not sufficient to find equilibrium values. of spin modules and the quadrupole matrix. Therefore, we added half-Casimirs s 2 and s 4 , which have a lower SO(3) symmetry, to the expression (14). We believe that these terms are small, so that SU(3) symmetry properties of the homogeneous part of the exchange energy are approximately conserved. In case of the SO(3) symmetric exchange interaction, the energy expression transforms to the known form of the exchange energy of a magnet with the spin s=1/2. The explicit form of the homogeneous part of the exchange energy (14) makes it possible to find equilibrium values of magnetic parameters and regions of existence of magnetic phases. In case of the degenerate multisublattice magnet, when choosing a homogeneous part of the exchange energy, we confine ourselves to its dependence on the Casimir invariant a 2 of the extended algebra (3), (4), and also half-Casimirs:. g 2 of the Poisson bracket subalgebra (3); n 2 of the Poisson bracket subalgebra (6), (8); s 2 of the Poisson bracket. . . subalgebra (6). Thus, the homogeneous exchange energy can be represented as follows: e0  e0(1) g 2 , a2   e0( 2) s 2 , n 2 . We assume the additional term of the. e0( 2). s , n  form to be small, so that it does not affect dynamic equations, but it 2. 2. affects the stability of equilibrium states. The term e0(1) g 2 , a2  is SU(3) symmetric, and in case of the presence of only the SU(3) symmetric Hamiltonian, the quantity e0 / g 2.  0 is in equilibrium. The inclusion of SO(3) symmetric 0 terms in the energy model representation makes it possible to obtain equilibrium values at e0 / g 2  0 , which leads 0 to new branches of magnetic excitation the spectra. DYNAMIC EQUATIONS AND SPECTRA OF COLLECTIVE EXCITATIONS Relations (1),(3),(4) allow us to obtain dynamic equations and find spectra of collective excitations of degenerate magnets with the spin s=1. To construct dynamic equations, the inhomogeneous part of the exchange energy be chosen in the following form, according to [23]. en  JSp( k gˆ ) 2 / 2,. en  JSp k gˆ  / 2  JSp k aˆ  / 2. 2. 2. (15). Here, J , J are constants of the inhomogeneous exchange interaction. The first formula in (15) corresponds to a singlesublattice magnet, and the second one corresponds to a multisublattice magnet. From here, we obtain dynamic equations of single-sublattice magnets with the SU(3) symmetry: (16) gˆ x  iJ gˆ x, gˆ x. Stable equilibrium values of the quantities q , s are local minimum points of the function e0 q, s  (14). From the conditions e0 / s  0, e0 / q  0 we find minimum energy point qi0 , si0 . Then we linearize the equation (16) near these equilibrium states and write out the spectra:.

(9) 8. EEJP Vol.4 No.2 2017. A.V. Glushchenko, M.Y. Kovalevsky. 1. The solution s0  q0  0 corresponds to a stable paramagnetic state if exchange interaction constants satisfy the inequalities J 0  0 and A  J 0  0 . The spectra in this case are degenerate. J0  A , q0  0 describes the ferromagnetic equilibrium state. For its stability, it is necessary to BC satisfy the inequalities: J 0  A  0, B  C  0 , J 0C  BA  0 . The spectra of collective excitations have the following. 2. The solution s 02 . form:   Jk 2 s0 ,  . 1 2 Jk s0 . 2. 3J 0 characterizes the quadrupole magnetic state, the stability of which is ensured by 4B. 3. The solution s0  0 , q0  2. inequalities J 0  0, A  0 , B  0 . We obtained the spectrum of magnetic excitations:   Jk 2 q0 .. 3 J 0C  BA A 2 , q0  determines the ferro-quadrupole ordering of the magnetic medium. This 4 CB C   solution is stable if: J 0C  BA  0, A  0 , B  0 , C  0 . The spectra of magnetic excitations are given by: s 0 || e0 :   1 1 2 2   Jk 2 s0 ,   q0  s0 Jk 2 , and s0  e0 :   q0  s0 Jk 2 ,    q0 2  s0 2  q0  Jk 2 . 2 2  We consider degenerate states of a multi-sublattice magnet. Using the Poisson brackets (3),(4), we find dynamic equations:  eˆgˆ , aˆ   eˆgˆ , aˆ   Hˆ gˆ , aˆ      k i aˆ , , gˆ x    k i  gˆ , (17) aˆ ( x)  i  , aˆ x .   k g    k a   g x   4. The solution s02  . . . ˆ , ex  0. Taking into account The SU(3) symmetry of the exchange interaction energy density is considered here G formulas (3),(4), we write out dynamic equations of a degenerate magnet with the spin s=1 in the multisublattice case,. . using the homogeneous energy structure e0  e0(1) g 2 , a2   e0( 2) s 2 , n 2 energy part (15):.  and the explicit form of the inhomogeneous. e aˆ, gˆ   iJ aˆ, gˆ . gˆ  iJ gˆ , gˆ   iJ aˆ, aˆ ; aˆ  i g 2 Next, we linearize equations (18) around possible equilibrium states.   ˆ 0  0 . Under condition e0 || f 0 , the spectra take the form: 1. Quadro-nematic qˆ 0  0 , m.     P  Jk q .    P0  Jk 2 q0 / 2  2. 0. Here P0  e0 / g 2. 0. 0. /2. P  Jk Jk q P  Jk Jk q.   P q  / 2.. 2. 2 2 0.  4 Jk 2 m02  P0 q02 / 2 ,. 2. 2 2 0.  4 Jk 2 m02. 0. 0.  . In case e0  f 0 , there are additional solutions:. . (18). 2 0 0. .   Jq 0 k 2 ,   P0 k 2 ,   J P0  Jk 2 m0 k ..   2. Quadro-antiferromagnet qˆ0  0 , nˆ0  0 . In this case, the spectra of collective excitations take the form at e0 || n0 :. .   In case e0  n0 :. P  Jk J n k , 2. 0. 0. . P  Jk Jk 2. 0. 2 2 q0. . . .  Jk 2 n02  P0 q02 / 2  P0  Jk 2 q0 / 2 ,.   JJ n0 k 2 / 2 ,   J 2 q02  4n02 JJ k 2 ,.   JJk 2  F0 q02 J n0 k ,    J 2q02  4n02 JJ  Jq0 k 2 / 2,   where F0   2 e0 / g 22 . In this case, along with the quadratic spectra, we obtain the Goldstone spectra for small values 0  of the wave vector k , which were previously predicted in work [25].. . . а) Antiferro-nematic gˆ 0  0 , aˆ 0  0 . The spectra of collective excitations have the form at f 0 ||n0 :. . . when e  n :.   JP0  JJk 2 n0 k ,.   n0  2m0 JP0  JJk 2 n0 k ,.

(10) 9. Models of Hamiltonian and Low-Frequency Spectra of Collective Excitations.... . n. 2 0. . .  m02 JP0  JJk 2 k ,. EEJP Vol.4 No.2 2017. . .    n02  2m02  2 m04  m02 n02  JP0  JJk 2 k ,  . where quantity n02  2m02  2 m04  m02 n02  0 for any values of m0 and n0 .   3. Ferro-quadromagnet gˆ 0  0 , aˆ 0  0 . From here we get the expression at s0 ||e0 :.   at s0  e0 :.   Jk 2 s0 ,   s0  2q0 Jk 2 / 2 ,   P0 s0 ,   s0  2q0 P0 / 2 ,.   q02  s02 k 2 J ,    q02  s02  q0 k 2 J / 2 ,    q02  s02  q0  P0 / 2 ,   q02  s02 P0 .     The comparison with the previously found spectra of collective excitations of the normal case leads to the observation that these spectra acquire an activation nature. It is easily seen that in case P0  0 , and when q0  0 or s0  0 , the spectra of magnetic excitations become the known results of [12]. CONCLUSIONS We have considered single-sublattice and multisublattice degenerate states of magnets with the spin s=1. We have obtained spectra of collective excitations and proposed an explicit form of the energy model presented in terms of Casimir invariants. For the homogeneous part of the exchange energy, we have found conditions for the existence of local minima, which correspond to equilibrium values of the magnet. In this paper, we have investigated a number of new magnetic states. They include the ferro-quadrupole state, quadro-nematic, quadro-antiferromagnetic, and antiferro-nematic state of the magnetic medium. In case of degenerate magnetic systems, the form of the homogeneous energy model affects the stability of equilibrium states and spectra of collective excitations. In contrast to Bogolyubov’s approach [14], where model representations of the energy are not required, the issue of choosing the density of the homogeneous energy takes one of the key values in the phenological approach. The presence of the set of Casimir invariants makes it possible to reduce the number of degrees of freedom and expand possibilities of the model representation of the exchange energy. The issue of finding the complete set of functionally independent Casimir invariants for degenerate magnetic media with the spin s>1/2 remains unsolved. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.. REFERENCES Demishev S.V. et al. Antiferro-quadrupole resonance in CeB 6 // Physica B: Condensed Matter. – 2006. – Vol. 378. – P. 602603. Takeya H. et al. Spin dynamics and spin freezing behavior in the two-dimensional antiferromagnet NiGa2S4 revealed by GaNMR, NQR and μSR measurements // Physical Review B. – 2008. – Vol. 77. – No. 5. – P. 054429. Santini P. et al. 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(12) 11. EAST EUROPEAN JOURNAL OF PHYSICS East Eur. J. Phys. Vol.4 No.2 (2017) 11-18. DOI:10.26565/2312-4334-2017-2-02. PACS: 29.17.+w; 41.75.Lx. INCREASE OF AMPLITUDE OF ACCELERATING WAKEFIELD EXCITED BY SEQUENCE OF SHORT RELATIVISTIC ELECTRON BUNCHES IN PLASMA AT MAGNETIC FIELD USE D.S. Bondar1, I.P. Levchuk2, V.I. Maslov2, I.N. Onishchenko2 1. Karazin Kharkiv National University 61022, Kharkiv, Ukraine 2 NSC Kharkov Institute of Physics & Technology 61108 Kharkiv, Ukraine e-mail: [email protected] Received April 11, 2017 Earlier, the authors found a mechanism for the sequence of short relativistic electron bunches, which leads to resonant excitation of the wakefield, even if the repetition frequency of bunches differs from the plasma frequency. In this case, the synchronization of frequencies is restored due to defocusing of the bunches which get into the bad phases with respect to the plasma wave. However, in this case, the bunches are lost, which as a result of this do not participate in the excitation of the wakefield. In this paper, numerical simulation was used to study the dynamics of electron bunches and the excitation of the wakefield in a magnetized plasma by a long sequence of short bunches of relativistic electrons. When a magnetic field is used, the defocussed bunches return to the region of interaction with the field after a certain time. In this case, the electrons of the bunches, returning to the necessary phases of the field, participate in the excitation of the wakefield. Also, the use of a magnetic field leads to an increase of the frequency of the excited wave relative to the repetition frequency of bunches. The latter increases the time for maintaining the resonance and, consequently, leads to an increase of the amplitude of the excited wakefield. KEYWORDS: wakefield, relativistic electrons, maintenance of the existence of a resonance, an increase of the amplitude of the excited wakefield. ЗБІЛЬШЕННЯ АМПЛІТУДИ ПРИСКОРЮЮЧОГО КІЛЬВАТЕРНОГО ПОЛЯ, ЯКЕ ЗБУДЖУЄТЬСЯ ПОСЛІДОВНІСТЮ КОРОТКИХ РЕЛЯТИВІСТСЬКИХ ЕЛЕКТРОННИХ ЗГУСТКІВ В ПЛАЗМІ, ПРИ ВИКОРИСТАННІ МАГНІТНОГО ПОЛЯ Д.С. Бондарь1, І.П. Левчук2, В.І. Маслов2, І.М. Онищенко2 1 Харківський національний університет імені В.Н. Каразіна 61022, Харків, Україна 2 ННЦ Харківський фізико-технічний інститут 61108, Харків, Україна Раніше авторами був знайдений для послідовності коротких релятивістських електронних згустків механізм, який призводить до резонансного збудження кільватерного поля, навіть якщо частота проходження згустків відрізняється від плазмової частоти. В цьому випадку синхронізація частот відновлюється за рахунок дефокусування згустків, які потрапляють в погані фази по відношенню до плазмової хвилі. Однак при цьому втрачаються згустки, які в результаті цього не беруть участі в збудженні кільватерного поля. У цій роботі чисельним моделюванням вивчена динаміка електронних згустків і збудження кільватерного поля в замагніченій плазмі довгою послідовністю коротких згустків релятивістських електронів. При використанні магнітного поля дефокусовані згустки через певний час повертаються в область взаємодії з полем. При цьому електрони згустків, що повертаються в потрібні фази поля, беруть участь в збудженні кільватерного поля. Також використання магнітного поля призводить до збільшення частоти збуджуваної хвилі щодо частоти проходження згустків. Останнє збільшує час підтримки резонансу і, отже, призводить до збільшення амплітуди кільватерного поля. КЛЮЧОВІ СЛОВА: кільватерне поле, релятивістські електрони, підтримання існування резонансу, збільшення амплітуди збуджуваного кільватерного поля. УВЕЛИЧЕНИЕ АМПЛИТУДЫ УСКОРЯЮЩЕГО КИЛЬВАТЕРНОГО ПОЛЯ, ВОЗБУЖДАЕМОГО ПОСЛЕДОВАТЕЛЬНОСТЬЮ КОРОТКИХ РЕЛЯТИВИСТСКИХ ЭЛЕКТРОННЫХ СГУСТКОВ В ПЛАЗМЕ, ПРИ ИСПОЛЬЗОВАНИИ МАГНИТНОГО ПОЛЯ Д.С. Бондарь1, И.П. Левчук2, В.И. Маслов2, И.Н. Онищенко2 1 Харьковский национальный университет имени В. Н. Каразина 61022, Харьков, Украина 2 ННЦ Харьковский физико-технический институт 61108, Харьков, Украина Ранее авторами был найден для последовательности коротких релятивистских электронных сгустков механизм, который приводит к резонансному возбуждению кильватерного поля, даже если частота следования сгустков отличается от плазменной частоты. В этом случае синхронизация частот восстанавливается за счет дефокусировки сгустков, которые попадают в плохие фазы по отношению к плазменной волне. Однако при этом теряются сгустки, которые в результате этого не участвуют в возбуждении кильватерного поля. В этой работе численным моделированием изучена динамика электронных сгустков и возбуждение кильватерного поля в замагниченной плазме длинной последовательностью коротких сгустков релятивистских электронов. При использовании магнитного поля дефокусированные сгустки через определенное время возвращаются в область взаимодействия с полем. При этом электроны сгустков, возвращающиеся в нужные фазы. © Bondar D.S., Levchuk I.P., Maslov V.I., Onishchenko I.N., 2017.

(13) 12. EEJP Vol.4 No.2 2017. D.S. Bondar, I.P. Levchuk.... поля, участвуют в возбуждении кильватерного поля. Также использование магнитного поля приводит к увеличению частоты возбуждаемой волны относительно частоты следования сгустков. Последнее увеличивает время поддержания резонанса и, следовательно, приводит к увеличению амплитуды возбуждаемого кильватерного поля. КЛЮЧЕВЫЕ СЛОВА: кильватерное поле, релятивистские электроны, поддержание существования резонанса, увеличению амплитуды возбуждаемого кильватерного поля. Development of accelerators of charged particles, in particular, colliders is one of the most promising areas of current research. Modern accelerators on metal structures are huge and expensive, because metal breaks out at 100MeV/m. It is clear that to achieve in linear colliders high energy (above 1 TeV) it is necessary to increase their length to tens of kilometers. One can estimate which dimensions of the accelerators should be used to accelerate electrons to the required energy of 1 TeV. It's tens of kilometers. It is necessary to make them smaller and cheaper. To do this, it is necessary to increase the rate of acceleration, i.e. to increase the accelerating field. In a plasma, one can excite the field Ez=100 GeV/m. This can be done using a wakefield, excited by a bunch of electrons, by a bunch of ions or by a laser pulse. In the experiment, the record results were already obtained: the laser pulse accelerated electrons in the plasma to 4.2 GeV at a distance 9cm. Therefore, the electric field equals Ez≈47 GeV/m [1]. Also in an experiment in plasma, a dense electron bunch with the energy of 42 GeV excited the wakefield, and its tail accelerated to the energy 84 GeV (i.e., doubled the energy) at a distance of approximately 1 m [2], then the electric field is Ez≈42 GeV/m. In a dielectric in the experiment, an impulse field of approximately 10 GeV/m has been excited. I.e. the dielectric accelerator can be in 100 times shorter than the metallic accelerator, and the plasma accelerator is in 1000 times shorter. Because the dielectric accelerator is easier to operate, and the plasma provides larger fields, the dielectric and plasma accelerators are intensively investigated. The wakefield excitation in a plasma by a long sequence of electron bunches is considered in this paper. An important factor of interest to this case is the use of external longitudinal (along the axis) magnetic field with the purpose to increase the accelerating field. So, the aim of the work is to investigate the features of use some optimal magnetic field, that would ensure a greater growth of the rate of acceleration. PROBLEM STATEMENT At resonant excitation of the wakefield in the plasma, the repetition frequency of bunches ωm , is equal to the frequency of the excited wakefield ωpe = ωm , i.e. to electron plasma frequency ωpe . Since along the radius r, the wakefield is localized near the sequence of bunches, i.e. along the radius r, it is localized in some neighborhood of the sequence of bunches, then the wakefield has not only a longitudinal Ez accelerating/decelerating field, but also a radial Fr focusing/defocusing force. The field Ez and Fr are shifted relative to each other by a quarter of the wavelength, i.e. on π/2. Where Ez=Ez max, there Fr=0; and where Ez=0, there Fr=Fr max. Since the bunches are finite size, the part of the bunch is defocused and ceases to excite wakefield. Part firstly is focused, and then due to the expansion of betatron oscillations (i.e., radial in the radial potential well) is again defocused, i.e. it leaves along r the region of interaction with the field and ceases to excite the wakefield too. Therefore, it is advisable not to allow the bunches to be defocused or periodically return them by an external longitudinal (along the axis) magnetic field. Although it is known that the use of a magnetic field in the experiment is associated with additional difficulties. But the magnetic field suppresses the focusing, and the maximum wakefield has been observed when the bunches are focused by the wakefield. Also, the magnetic field suppresses defocusing. And it is very important for the excitation of the wakefield, because in the experiment it is very difficult to maintain the resonant plasma because of its uncontrolled inhomogeneity and nonstationarity. And in the nonresonant case only small amplitude beatings are excited. However, it was shown in [3-7] that due to the self-cleaning of the "bad" bunches at their defocusing and in the nonresonant case, intense excitation of the wakefield is possible. ANALYSIS OF THE USING OF A MAGNETIC FIELD As a consequence, it would seem that the use of a magnetic field is impractical, it suppresses defocusing. Hence it follows that one must use some optimal magnetic field when it does not yet suppress focusing and defocusing, but it already returns defocussed bunches after some time into the region of interaction with the field. Then it is necessary to use a not very strong optimal magnetic field, so that after defocusing the bunches they return to the axis after some time, and they again excite the wakefield. The return time is 2π/ωce. Where ωce is the electron cyclotron frequency. In order that H0 allows electrons to reach the axis in the focusing field, the radius of the radial oscillations of electrons in crossed H0z and Er fields should be not less than the radius of the bunch eE r > rb . m e γ b ωce2. (1). It is necessary that for N-th bunch (if N bunches excite the maximum wakefield) it is satisfied when the maximum wakefield is reached. Sequential bunches excite wakefield that firstly it grows linearly with increasing number of injected bunches.

(14) Increase of Amplitude of Accelerating Wakefield, Excited by Sequence of Short.... 13. EEJP Vol.4 No.2 2017. E rN = NE r1 .. ErN is the radial wakefield after N bunches. Er1 is the radial wakefield after the 1st bunch. Then it must be fulfilled eNE r1 > rb . m e γ b ωce2. (2). (3). It is necessary to note that an additional frequency shift occurs when a magnetic field is used which, at a certain value of the magnetic field, tightens the maintenance of the resonance, leading to an increase of the excited wakefield. A long sequence of electron bunches of a small charge was used in the experiment [8] and in numerical simulation [9] for excitation of an intense wakefield in a plasma. However, as it turned out, there is the limiting amplitude of the wakefield. It is determined by the fact that the nonlinear shift of the frequency of the wakefield appears with increase of the wakefield amplitude. Because of this, the resonant interaction of bunches with the wakefield is detuned. This resonance detuning is delayed in time, if, as it was done in [10], a small excess of the plasma density above the resonant value is chosen initially. The same resonance maintenance can be ensured by using a small magnetic field. Using the LCODE [11], numerical simulation of the growth of the wakefield amplitude was performed. It is shown that when the resonance is maintained, the amplitude of the wakefield increases in comparison with the case of the initial resonant conditions. The wakefield excitation in a plasma by a long sequence of electron bunches is considered in this paper. The resonant excitation of the wakefield by a long sequence of relativistic electron bunches is difficult, because it is difficult to maintain a homogeneous and stationary plasma in the experiment. However, intense wakefield excitation by a long sequence of relativistic electron bunches has been observed. The mechanism of resonant excitation of the wakefield by a nonresonant sequence of short electron bunches has been investigated in [9]. Frequency synchronization is carried out due to self-cleaning of the sequence of bunches due to defocusing and leaving along radius of some bunches that are not in phase with the wave. The simulation results of the mechanism of maintaining the resonance of electron bunches with a wakefield using a magnetic field are presented in this paper. In [9], for a sequence of short relativistic electron bunches, a mechanism was found which leads to a resonant excitation of the wakefield, even if the repetition frequency of bunches differs appreciably from the plasma frequency. Synchronization of frequencies is restored due to defocusing of bunches which get into bad phases with respect to the plasma wave. However, the bunches are lost, which as a result do not participate in the excitation of the wakefield. When a magnetic field is used, the defocused bunches return to the region of interaction with the field after a certain time. In this case, the electrons of the bunches returning to the necessary phases of the field can participate in the excitation of the wakefield. Also, the use of a magnetic field leads to an increase of the frequency of the excited wave relative to the repetition frequency of bunches. The latter increases the time for maintaining the resonance and, consequently, leads to an increase of the amplitude of the excited wakefield. SIMULATION OF RESONANCE RECOVERY FOR THE CASE OF 32 BUNCHES For numerical simulation parameters are selected: nres=1011cm-3 is the resonant plasma density which corresponds to ratio ωpe=ωm=2π·2.8·109, relativistic factor of bunches equals γb=5, have been selected. Where ωm is the repetition frequency of bunches, ωpe=(4πnrese2/me)1/2 is the electron plasma frequency. The density of bunches n b = 6 × 108 cm −3 is distributed in the transverse direction approximately according to Gaussian distribution, σ r = 0.5cm , λ = 10.55cm is the wavelength, ξ=Vbt-z, Vb is the velocity of bunches. Time is normalized on ωpe-1, distance - on c/ωpe, density - on nres, current Ib - on Icr=πmc3/4e, fields – on (4πnresc2me)1/2. We consider the dynamics of the first 32 bunches in plasma. We use the cylindrical coordinate system (r, z) and draw the plasma and beam densities at some z as a function of the dimensionless time τ=ωpt. The longitudinal coordinate ξ=z-Vbt is normalized on 2π/λ (λ is the wavelength). The values of the Ez, Fr, Hθ and H0 are normalized on mcωpe/e. Where e, m are the charge and mass of the electron, c is the light velocity, ωpe is the electron plasma frequency. We do not take into account the longitudinal dynamics of the bunches, because at the times and energies of the beam according to. dVz (r ) 1 dVr (r ) 1 ∝ 3 , ∝ dr γb dr γb. radial relative shifts of beam particles predominate. Vz, Vr are the longitudinal and radial velocities of the electron bunches, γb is the relativistic factor of the bunches. The wakefield excitation by 32 bunches is considered for two cases: initially the resonant case and the case of resonance recovery. We consider a sequence of 32 bunches which are uniform in the longitudinal direction and are.

(15) 14. EEJP Vol.4 No.2 2017. D.S. Bondar, I.P. Levchuk.... distributed according to Gaussian along the radius. The lengths of the bunches are chosen equal to half of the wavelength ξb=λ/2.. Fig. 1. Spatial distribution of density nb of sequence of initially radially Gaussian and longitudinally homogeneous resonant bunches into the plasma at ξb=λ/2, Ib=0.2×10-3. Fig. 1a (part of Fig. 1). Spatial distribution of density nb of 32nd initially radially Gaussian and longitudinally homogeneous resonant bunch into the plasma at ξb=λ/2, Ib=0.2×10-3. We consider the case when the initial density of plasma electrons n0 is such that the repetition frequency of the bunches ωm is equal to the electron plasma frequency ωm=ωpe. In this case, up to the point of maximum focusing of the bunches Ez grows and then it decreases during refocusing. To compensate the charge of the bunches, some of the plasma electrons leave the axis. Then the phase velocity of the wave Vph(r=0) (and ωpe(r=0)) on the axis becomes smaller than at the periphery with respect to r. As a result, the wave becomes skewed (Fig. 2.).. Fig. 2. Spatial distribution of plasma electron density ne in wakefield, excited by sequence of initially radially Gaussian and longitudinally homogeneous resonant bunches. Fig. 2a (part of Fig. 2). Spatial distribution of plasma electron density ne in 12th wavelength of wakefield, excited by sequence of initially radially Gaussian and longitudinally homogeneous resonant bunches.

(16) Increase of Amplitude of Accelerating Wakefield, Excited by Sequence of Short.... 15. EEJP Vol.4 No.2 2017. Also, in this case, Vph(r=0)<Vb is smaller than the beam velocity Vb, and ωpe(r=0)<ωm, i.e. the wave becomes nonresonant with a sequence of bunches. The wave lags behind the bunches and the smaller their parts get into the focusing fields (Figs. 1, 1a, 3, 3a, 5, 5a, 7, 7a).. Fig. 3. Longitudinal distribution of radius rb, of density nb of sequence of initially radially Gaussian and longitudinally homogeneous resonant bunches and of radial wake force Fr into the plasma at ξb=λ/2, Ib=0.2×10-3. Fig. 3a (part of Fig. 3). Longitudinal distribution of radius rb , of density nb of 32nd initially radially Gaussian and longitudinally homogeneous resonant bunch and of radial wake force Fr in 32nd wavelength of wakefield into the plasma at ξb=λ/2, Ib=0.2×10-3. In this case, a very small part of their first front gets into the accelerating phases (Figs. 4, 4a.).. Fig. 4. Longitudinal momenta of 32 bunches. Fig. 4a (part of Fig. 4). Longitudinal momenta of 30th bunch.

(17) 16. EEJP Vol.4 No.2 2017. D.S. Bondar, I.P. Levchuk.... Fig. 5. Spatial distribution of density nb of sequence of initially radially Gaussian and longitudinally homogeneous bunches into the plasma at ξb=λ/2, Ib=0.2×10-3, H0=0.1, ne/nres-1=0.02, nres is the resonant plasma electron density. ← Fig. 5a (part of Fig. 5). Spatial distribution of density nb of 32nd initially radially Gaussian and longitudinally homogeneous resonant (right) (left at H0=0.1, ne/nres-1=0.02) bunch into the plasma at ξb=λ/2, Ib=0.2×10-3. If we use a somewhat larger n0 and a small magnetic field H0=0.1, then the resonance of the wave with the bunch sequence is restored on the axis, since on the axis ωpe (0) ≈ ωm. However, at the periphery with respect to r, the wave becomes nonresonant with the bunch sequence, so now ωpe(r>0) > ωm. Then the wave becomes skewed in the opposite direction (Figs. 2, 2a, 6, 6a.).. Fig. 6. Spatial distribution of plasma electron density ne in wakefield, excited by sequence of initially radially Gaussian and longitudinally homogeneous bunches into the plasma at ξb=λ/2, Ib=0.2×10-3, H0=0.1, ne/nres-1=0.02. ← Fig. 6a (part of Fig. 6). Spatial distribution of plasma electron density ne in wakefield, excited by sequence of initially radially Gaussian and longitudinally homogeneous resonant (rigth) (left at H0=0.1, ne/nres-1=0.02) bunch into the plasma at ξb=λ/2, Ib=0.2×10-3, H0=0.1, ne/nres-1=0.02. When the resonance of a wave is restored to a sequence of bunches on the axis, larger parts of the bunches get into the focusing fields (Figs. 1, 1a, 3, 3a, 5, 5a, 7, 7a), and defocused bunches is weakly defocused (Fig. 7a),.

(18) Increase of Amplitude of Accelerating Wakefield, Excited by Sequence of Short.... 17. EEJP Vol.4 No.2 2017. Fig. 7. Longitudinal distribution of radius rb of density nb of sequence of initially radially Gaussian and longitudinally homogeneous bunches and of radial wake force Fr into the plasma at ξb=λ/2, Ib=0.2×10-3, H0=0.1, ne/nres-1=0.02. Fig. 7a (part of Fig. 7). Longitudinal distribution of radius rb of density nb of bunch and of radial wake force Fr into the plasma at H0=0.1, ne/nres-1=0.02). the excited wakefield becomes larger (Figs. 3,7) compared with the initially resonant and subsequent disturbance of the resonance, and small parts of the back fronts of the bunches get into the accelerating phases (Figs. 8).. Fig. 8. Longitudinal momenta of 32 bunches at ξb=λ/2, Ib=0.2×10-3, H0=0.1, ne/nres-1=0.02. At H0=0.1 the wakefield Ez became more uniform along the plasma, because it does not so rapidly decrease to the end of the plasma. CONCLUSION The main conclusion is that, magnetic field can be used for wakefield increase and for increase the transformation of driver-bunch energy into the accelerated electrons (witness) energy. In this paper it is shown that in order to improve the energy transformation of driver-bunch energy into the witness energy, some optimal magnetic field should be used when it does not yet suppress focusing and defocusing. Such optimal magnetic field should also ensure returns defocussed bunches after some time into the region of the interaction with the field. Moreover, it is important to choose such optimal magnetic field that would return the bunches to the axis, after some time that bunches would again excite the wakefield. It is also important to note that the use of a magnetic field leads to an increase in the frequency of the excited wave relative to the repetition frequency of bunches. Using the code lcode [9], numerical simulation of the growth of the wakefield amplitude was performed. It is shown that when the resonance is maintained, the amplitude of the wakefield increases in comparison with the case of the initial resonant conditions. The simulation results of the mechanism of maintaining the resonance of electron bunches with a wakefield using a magnetic field are presented in this paper. 1.. REFERENCES Leemans W.P., Gonsalves A.J., Mao H.-S. et al. Multi-GeV Electron Beams from Capillary-Discharge-Guided Subpetawatt Laser Pulses in the Self-Trapping Regime // Phys. Rev. Lett. – 2014. – Vol. 113. - P. 245002..

(19) 18. EEJP Vol.4 No.2 2017 2.. D.S. Bondar, I.P. Levchuk.... Blumenfeld I., Clayton C.E., Decker F.-J., et al. Energy doubling of 42 GeV electrons in a metre-scale plasma wakefield accelerator // Nature, Letters. -2007. - Vol. 445. - P. 741-744. 3. Lotov K.V, Maslov V.I., Onishchenko I.N., Svistun E. Resonant excitation of plasma wakefields by a nonresonant train of short electron bunches // Plasma Phys. Control. Fusion. – 2010. – Vol.52 . – No.6. – P.065009. 4. Lotov K.V, Maslov V.I., Onishchenko I.N., Svistun E. 2.5D simulation of plasma wakefield excitation by a nonresonant chain of relativistic electron bunches // Problems of Atomic Science and Technology. – 2010. – No.2. – P.122-124. 5. Lotov K. V., Maslov V. I., Onishchenko I. N., Svistun O. M., Vesnovskaya M.S. To the Plasma Wakefield Excitation by a Nonresonant Sequence of Relativistic Electron Bunches at Plasma Frequency above Bunch Repetition Frequency // Problems of Atomic Science and Technology. Ser. Plasma Physics. – 2010. – No.6. . – P.114 – 116. 6. Lotov K. V., Maslov V. I., Onishchenko I. N. Long Sequence of Relativistic Electron Bunches as a Driver in Wakefield Method of Charged Particles Acceleration in Plasma // Problems of Atomic Science and Technology. Ser. Plasma Physics. – 2010. – No.6. . – P.103 – 107. 7. Lotov K.V, Maslov V.I., Onishchenko I.N., Yarovaya I.P. Mechanisms of Synchronization of Relativistic Electron Bunches at Wakefield Excitation in Plasma // Problems of Atomic Science and Technology. – 2013. –Vol.–86.– No.4. – P.73-76. 8. Berezin А.К., Fainberg Ya.B., Kiselev V.A., et al., Wakefield excitation in plasma by relativistic electron beam, consisting regular train of short bunches // Phys. Plasmas. – 1994–Vol.–20. – No.7. – P.663-670. 9. Lotov K.V, Maslov V.I., Onishchenko I.N., Svistun E.N. Resonant excitation of plasma wakefields by a non-resonant train of short electron bunches // Plasma Phys. Control. Fusion. – 2010–Vol.–52. – P.065009. 10. Lotov K.V, Maslov V.I., Onishchenko I.N., Svistun E. N., 2.5D simulation of plasma wakefield excitation by a nonresonant chain of relativistic electron bunches // Problems of Atomic Science and Technology. – 2010. – No.– P.122-124. 11. Lotov K.V. Simulation of ultrarelativistic beam dynamics in plasma wakefield accelerator // Phys. Plasmas. – 1998. – Vol.5. – No.3.– P.785-791..

(20) 19. EAST EUROPEAN JOURNAL OF PHYSICS East Eur. J. Phys. Vol.4 No.2 (2017) 19-28. DOI:10.26565/2312-4334-2017-2-03. PACS: 87.14.Cc, 87.16.Dg. EFFECT OF AMYLOID FIBRILS ON ELECTROKINETIC PROPERTIES OF LIPID VESICLES. 1. U. Tarabara1, K. Vus1, S. Girnyk1, N. Kamneva2, O. Lavryk1, M. Mikhailyuta1, V. Trusova1, G. Gorbenko1 Department of Nuclear and Medical Physics, V.N. Karazin Kharkiv National University 2 Department of Physical Chemistry, V.N. Karazin Kharkiv National University 4 Svobody Sq., Kharkiv, 61022, Ukraine e-mail: [email protected] Received April 1, 2017. The influence of the lysozyme and serum albumin in their native and amyloid forms on the electrokinetic behavior of the negatively charged uni- and multilamellar liposomes from the zwitterionic lipid phosphatidylcholine and anionic lipid cardiolipin has been investigated using the microelectrophoresis technique. The zeta - potential, the surface electrostatic potential and surface charge density of the lipid vesicles have been determined upon varying the lipid-to-protein molar ratio. The complex dependencies of the electrophoretic mobility on the protein concentration and reversal of the surface charge observed for the multilamellar vesicles have been explained by the multilayer protein adsorption on the liposomal surface. It has been found that the native and fibrillar proteins differ in their ability to modify the charge state of the model membranes. KEYWORDS: electrophoretic mobility, lipid vesicles, lysozyme, serum albumin, amyloid fibrils ВЛИЯНИЕ АМИЛОИДНЫХ ФИБРИЛЛ НА ЭЛЕКТРОКИНЕТИЧЕСКИЕ СВОЙСТВА ЛИПИДНЫХ ВЕЗИКУЛ У. Тарабара1, Е. Вус1, С. Гирнык1, Н. Kaмнева2, O. Лаврик1, M. Михайлюта1, В. Трусова1, Г. Горбенко1 1 Кафедра ядерной и медицинской физики, Харьковский национальный университет имени В.Н. Каразина 2 Кафедра физической химии, Харьковский национальный университет имени В.Н. Каразина Харьковский национальный университет им. В.Н. Каразина пл. Свободы 4, Харьков, 61022, Украина Методом микроэлектрофореза исследовано влияние нативной и амилодной форм лизоцима и сывороточного альбумина на электрокинетическое поведение моно- и мультиламеллярных липосом из цвиттерионного липида фосфатидилхолина и анионного липида кардиолипина. При варьировании молярного соотношения липид:белок были определены дзета – потенциал, поверхностный электростатический потенциал и поверхностная плотность заряда липидных везикул. Сложная зависимость электрофоретической подвижности от концентрации белка и изменение знака поверхностного заряда, выявленные для мультиламеллярных везикул, были объяснены мультислойной адсорбцией белков на поверхности липосом. Обнаружено, что нативная и фибриллярная формы белков различаются по их способности модифицировать зарядовое состояние модельных мембран. КЛЮЧЕВЫЕ СЛОВА: электрофоретическая подвижность, липидные везикулы, лизоцим, сывороточный альбумин, амилоидные фибриллы ВПЛИВ АМІЛОЇДНИХ ФІБРИЛ НА ЕЛЕКТРОКІНЕТИЧНІ ВЛАСТИВОСТІ ЛІПІДНИХ ВЕЗИКУЛ У. Тарабара1, K. Вус1, С. Гірник1, Н. Kaмнєва2, O. Лаврик1, M. Михайлюта1, В. Трусова1, Г. Горбенко1 1 Кафедра ядерної та медичної фізики, Харківський національний університет імені В.Н. Каразіна 2 Кафедра фізичної хімії, Харківський національний університет імені В.Н. Каразіна пл. Свободи 4, Харків, 61022, Україна Методом мікроелектрофорезу досліджено вплив нативної та амілоїдної форм лізоциму та сироваткового альбуміну на електрокінетичну поведінку моно- и мультиламелярних ліпосом із цвіттеріонного ліпіду фосфатидилхоліну та аніонного ліпіду кардіоліпіну. При варіюванні молярного співвідношення ліпід:білок було визначено дзета – потенціал, поверхневий електростатичний потенціал та поверхневу густину заряду ліпідних везикул. Складна залежність електрофоретичної рухливості від концентрації білка та зміна знаку поверхневого заряду, виявлені для мультиламелярних везикул, було пояснено мультишаровою адсорбцією білків на поверхні ліпосом. Встановлено, що нативна та фібрилярна форми білків відрізняються за їх здатністю модифікувати зарядовий стан модельних мембран. КЛЮЧОВІ СЛОВА: електрофоретична рухливість, ліпідні везикули, лізоцим, сироватковий альбумін, амілоїдні фібрили. Electrostatic phenomena are known to play an essential role in determining the structural and functional properties of biological systems [1,2]. In particular, electrostatics controls a wide variety of processes occurring in cellular membranes, among which are non-specific and specific protein–lipid interactions, protein folding, translocation and orientation in the lipid bilayer [3-5], enzyme functioning , ion binding and transport, structural and phase transitions in the lipid phase [1], recognition events [6,7], pharmacological effects [8], etc. Biological membranes consist of hundreds of different molecular species including lipids, proteins and carbohydrates bearing numerous ionized groups which account for the net negative charge of the membrane surface [1]. An essential part of such groups belong to anionic lipids such as phosphatidylglycerol, cardiolipin, phosphatidylserine, phosphatidic acid and phosphatidylinositol, whose © Tarabara U., Vus K., Girnyk S., Kamneva N., Lavryk O., Mikhailyuta M., Trusova V., Gorbenko G., 2017.

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